R E V I E W
Open Access
Recent developments on the moment
problem
Gwo Dong Lin
Correspondence: [email protected] Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, Republic of China
Abstract
We consider univariate distributions with finite moments of all positive orders. The moment problem is to determine whether or not a given distribution is uniquely determined by the sequence of its moments. There is a huge literature on this classical topic. In this survey, we will focus only on the recent developments on thecheckable
moment-(in)determinacy criteria including Cramér’s condition, Carleman’s condition, Hardy’s condition, Krein’s condition and the growth rate of moments, which help us solve the problem more easily. Both Hamburger and Stieltjes cases are investigated. The former is concerned with distributions on the whole real line, while the latter deals only with distributions on the right half-line. Some new results or new simple (direct) proofs of previous criteria are provided. Finally, we review the most recent moment problem for products of independent random variables with different distributions, which occur naturally in stochastic modelling of complex random phenomena.
Keywords: Hamburger moment problem, Stieltjes moment problem, Cramér’s
condition, Carleman’s condition, Krein’s condition, Hardy’s condition
AMS Subject Classification: 60E05; 44A60
1 Introduction
The moment problem is a classical topic over one century old (Stieltjes 1894, 1895, Kjeldsen 1993, Fischer 2011, pp. 157–168). We start with the definition of the moment determinacy of distributions. LetX be a random variable with distributionF (denoted X ∼ F) and have finite momentsmk = E[Xk] for allk = 1, 2,. . .; namely, the absolute momentμk =E[|X|k]<∞for all positive integersk. IfFis uniquely determined by the sequence of its moments{mk}∞k=1, we say thatFis moment-determinate (in short,Fis M-det, orXis M-det); otherwise, we say thatFis moment-indeterminate (Fis M-indet, orX is M-indet).
The moment problem is to determine whether or not a given distributionFis M-det. Roughly speaking, there are two kinds of moment problems: Stieltjes (1894) moment problem deals with nonnegative random variables only, while Hamburger (1920, 1921) moment problem treats all random variables taking values in the whole real line.
We recall first two important facts:
Fact A. It is possible that a nonnegative random variableX is M-det in the Stieltjes sense, but M-indet in the Hamburger sense (Akhiezer 1965, p. 240). This happens only for somediscretenonnegative random variables with a positive mass at zero (Chihara 1968).
Fact B. If a distribution F is M-indet, then there are infinitely many (i) absolutely continuous distributions, (ii) purely discrete distributions and (iii) singular continu-ous distributions all having the same moment sequence as F (Berg 1998, Berg and Christensen 1981).
One good reason to study the moment problem was given in Frechet and Shohat’s´ (1931) Theorem stated below. Simply speaking, for a given sequence of random variables Xn ∼ Fn, n = 1, 2,. . ., with finite momentsm(kn) = E
Xk n
for all positive integersk,
the moment convergencelimn→∞m(kn)=mk∀k
does not guarantee the weak
conver-gence of distributions{Fn}∞n=1
Fn→w Fasn→ ∞
unless the limiting distributionFis M-det. Therefore, the M-(in)det property is one of the important fundamental properties we have to know about a given distribution.
Frechet and Shohat’s (1931) Theorem.´ Let the distribution functions Fn possess finite momentsm(kn) fork = 1, 2,. . .andn = 1, 2,. . .. Assume further that the limit mk = limn→∞m(kn)exists (and is finite) for eachk. Then (i) the limits{mk}∞k=1are the moment sequence of a distribution function, sayF; (ii) if the limitFgiven by (i) is M-det, Fnconverges toFweakly asn→ ∞.
Necessary and sufficient conditions for the M-det property of distributions exist in the literature (see, e.g., Akhiezer 1961, Shohat and Tamarkin 1943, and Berg et al. 2002), but these conditions are not easily checkable in general. In this survey, we will focus only on thecheckableM-(in)det criteria for distributions rather than the collection of all specific examples.
In Sections 2 and 3, we review respectively the moment determinacy and moment indeterminacy criteria including Cramér’s condition, Carleman’s condition, Hardy’s con-dition, Krein’s condition and the growth rate of moments. Some criteria are old, but others are recent. New (direct) proofs for some criteria are provided. To amend some previous proofs in the literature, two lemmas (Lemmas 3 and 4) are given for the first time. We consider in Section 4 the recently formulated Stieltjes classes for M-indet abso-lutely continuous distributions. Section 5 is devoted to the converses to the previous M-(in)det criteria for distributions. Finally, in Section 6 we review the most recent results about the moment problem for products of independent random variables with different distributions.
2 Checkable criteria for moment determinacy
In this section we consider the checkable criteria for moment determinacy of random variables or distributions. We treat first the Hamburger case because it is more popu-lar than the Stieltjes case. LetX ∼ Fon the whole real lineR = (−∞,∞)with finite momentsmk =E[Xk] and absolute momentμk = E[|X|k] for all positive integersk. For convenience, we define the following statements, in which ‘h’ stands for ‘Hamburger’.
(h1)m2(k+1)
m2k =O((k+1)
2)=O(k2)ask→ ∞.
(h2)Xhas a moment generating function (mgf ), i.e.,E[etX]<∞for allt∈(−c,c), where c>0 is a constant (Cramér’s condition); equivalently,E[et|X|]<∞for 0≤t<c. (h3) lim supk→∞21km12/(k2k)<∞.
(h4) lim supk→∞1kμ1k/k<∞. (h5)m2k=O
(2k)2kask→ ∞.
(h7)C[F]≡∞k=1m2−k1/(2k)= ∞(Carleman’s (1926) condition). (h8)Xis M-det onR.
Theorem 1Under the above settings, if X ∼F onRsatisfies one of the conditions (h1) through (h7), then X is M-det onR. Moreover, (h1) implies (h2), (h2) through (h6) are equivalent, and (h6) implies (h7). In other words, the following chain of implications holds:
(h1)=⇒(h2)⇐⇒(h3)⇐⇒(h4)⇐⇒(h5)⇐⇒(h6)=⇒(h7)=⇒(h8).
We keep the termk+1 in (h1) because it arises naturally in many examples. The first implication in Theorem 1 was given in Stoyanov et al. (2014) recently, while the rest, more or less, are known in the literature. The Carleman quantityC[F] in (h7) is calculated from all even order moments ofF. Theorem 1 contains most checkable criteria for moment determinacy in the Hamburger case.
Remark 1Some other M-det criteria exist in the literature, but they are seldom used. See, for example, (ha) and (hb) below:
(h2) X has a mgf (Cramér’s condition)
⇐⇒(ha)∞k=1 m2k
(2k)!x2kconverges in an interval|x|<x0(Chow and Teicher 1997, p. 301) =⇒(hb)∞k=1mkk!xkconverges in an interval|x|<x0(Billingsley 1995, p. 388)
=⇒(h7) C[F]=∞k=1m2−k1/(2k)= ∞(Carleman’s condition)
=⇒(h8) X is M-det onR.
It might look strange that the convergence of subseries in the above (ha) implies the con-vergence of the whole series in (hb), but remember that the concon-vergence in (ha) holds true for all x in a neighborhood of zero, not just for a fixed x.Billingsley (1995)proved the impli-cation that (hb)=⇒(h8) by a version of analytic continuation of characteristic function, but it is easy to see that (hb) also implies (h7) and hence X is M-det onR.
In Theorem 1, Carleman’s condition (h7) is the weakest checkable condition forXto be M-det onR. To prove Carleman’s criterion that (h7) implies (h8), we may apply the approach of quasi-analytic functions (Carleman 1926, Koosis 1988), or the approach of Lévy distance (Klebanov and Mkrtchyan 1985). For the latter, we recall the following result.
Klebanov and Mkrtchyan’s (1985) Theorem.LetF andGbe two distribution func-tions onRand let their first 2nmoments exist and coincide:mk(F) = mk(G) = mk, k=1, 2,. . ., 2n(n≥2). Denote the sub-quantityCn=nk=1m−
1/(2k)
2k . Then
L(F,G)≤c2
log(1+Cn−1) (Cn−1)1/4
,
where L(F,G) is the Lévy distance and c2 = c2(m2) depends only on m2. Therefore, Carleman’s condition (h7) implies thatF =Gby lettingn → ∞in Klebanov and Mkrtchyan’s (1985) Theorem. It worths mentioning that Carleman’s condition is suf-ficient, but not necessary, for a distribution to be M-det. For this, see Heyde (1963b), Stoyanov and Lin (2012, Remarks5and7) or Stoyanov (2013, Section 11).
it is easy to estimate the growth rate (see the example below), because the common fac-tors in the two even order moments,m2(k+1)andm2k, can be cancelled out asntends to infinity.
Example 1Consider the double generalized gamma random variableξ ∼DGG(α,β,γ ) with density function f(x)= c|x|γ−1exp[−α|x|β] , x∈R,whereα,β,γ >0,f(0) =0if γ =1, and c=βαγ /β/(2(γ /β))is the norming constant. Then the nth powerξnis M-det if1≤n≤β. To see this known result, we calculate the ratio of even order moments ofξn:
E[ξ2n(k+1)] E[ξ2nk] =
((γ +2n(k+1))/β)
α2n/β((γ +2nk)/β) ≈(2n/(αβ))2n/β(k+1)2n/β as k→ ∞,
by using the approximation of the gamma function:(x) ≈ √2πxx−1/2e−x as x → ∞. Therefore,ξnis M-det if n≤ β, by the criterion (h1). In fact, for odd integer n≥ 1,ξnis M-det iff n≤β,and for even integer n≥2,ξnis M-det iff n≤2β,regardless of parameter γ.For further results about this distribution and its extensions, seeLin and Huang (1997), Pakes et al. (2001)andPakes (2014, Theorem 8.3), as well as Examples3and5below.
Remark 2We give here a direct proof of the equivalence of statements (h2), (h3) (h5) and (h6). First, for any nonnegative X,we have the equivalence of the following four statements (to be shown later):
E[ec√X]<∞for some constant c>0
iff mk≤ck0(2k)! , k=1, 2,. . .,for some constant c0>0 ifflim supk→∞1kmk1/(2k)<∞
iff mk=Ok2kas k→ ∞.
Next, consider a general X withE[et|X|]< ∞for0 ≤ t < c, namely,E[et√|X|2
]< ∞for some constant t>0. Then the kth moment of|X|2is exactly the2kth moment of X and we have immediately the following equivalences (by taking|X|2as the above nonnegative X): (h2) X has a mgf
iff (h6) m2k≤ck0(2k)! , k=1, 2,. . .,for some constant c0>0 ifflim supk→∞1km2k1/(2k)<∞(iff (h3)lim supk→∞21km
1/(2k)
2k <∞) iff m2k =O(k2k)as k→ ∞(iff (h5) m2k =O((2k)2k)as k→ ∞).
We now present the checkable M-det criteria in the Stieltjes case. ConsiderX ∼Fon
R+ = [0,∞)with finite mk = μk = E[Xk] for all positive integers k, and define the following statements, in which ‘s’ stands for ‘Stieltjes’.
(s1)mk+1
mk =O((k+1)2)=O(k2)ask→ ∞.
(s2)√Xhas a mgf (Hardy’s condition), i.e.,E[ec√X]<∞for some constantc>0. (s3) lim supk→∞1km1k/(2k)<∞.
(s4)mk =O(k2k)ask→ ∞.
(s5)mk ≤ck0(2k)! , k=1, 2,. . ., for some constantc0>0. (s6)C[F]=∞k=1mk−1/(2k)= ∞(Carleman’s condition). (s7)Xis M-det onR+.
Theorem 2Under the above settings, if X ∼ F onR+satisfies one of the conditions (s1) through (s6), then X is M-det onR+. Moreover, (s1) implies (s2), (s2) through (s5) are equivalent, and (s5) implies (s6). In other words, the following chain of implications holds:
The first implication above was given in Lin and Stoyanov (2015). Note that the moment conditions here are in terms of moments of all positive (integer) orders, rather than even order moments as in the Hamburger case. For example, the statement (s1) means that the growth rate of all moments (not only for even order moments) is less than or equal to two. Like Theorem 1, Theorem 2 contains most checkable criteria for moment determinacy in the Stieltjes case. Hardy (1917, 1918) proved that (s2) implies (s7) by two different approaches. Surprisingly, Hardy’s criterion has been ignored for about one century since publication. The following new characteristic properties of (s2) are given in Stoyanov and Lin (2012), from which the equivalence of (s2) through (s5) follows immediately.
Lemma 1Let a be a positive constant and X be a nonnegative random variable. (i) IfE[exp(cXa)]<∞for some constant c>0, then mk ≤(k/a+1)ck0, k=1, 2,. . ., for some constant c0>0.
(ii) Conversely, if, in addition, a ≤ 1, and mk ≤ (k/a+1)ck0, k = 1, 2,. . .,for some constant c0>0,thenE[exp(cXa)]<∞for some constant c>0.
Corollary 1Let a∈(0, 1]and X ≥0. ThenE[exp(cXa)]<∞for some constant c> 0 iff mk ≤(k/a+1)c0k, k=1, 2,. . .,for some constant c0>0.
Lemma 2Let a be a positive constant and X be a nonnegative random variable. Then lim supk→∞1kmak/k <∞iff mk ≤(k/a+1)ck0, k=1, 2,. . ., for some constant c0>0.
Corollary 2Let a∈ (0, 1]and X ≥0.ThenE[exp(cXa)]<∞for some constant c> 0 ifflim supk→∞1kmak/k<∞.
We mention that for any nonnegativeX, its mgf exists iff lim supk→∞1km1k/k <∞due to Corollary 2. This in turn implies the equivalence of (h2) and (h4) in Theorem 1 for the Hamburger case. More general results in terms of absolute moments are given below. For easy comparison, some statements are repeated here.
Equivalence Theorem A(Hamburger case). Letp≥ 1 be a constant and the random variableX∼FonR. Denotemk =E[Xk] for integerk≥1 and letμ
=E[|X|]<∞for all >0. Then the following statements are equivalent:
(a)Xsatisfies Cramér’s condition, namely, the moment generating function ofXexists. (b)μk ≤ck0k! ,k=1, 2,. . ., for some constantc0>0.
(c)μpk ≤ck0(pk+1),k=1, 2,. . ., for some constantc0>0. (d) lim supk→∞pk1μ1pk/(pk)<∞.
(e)m2k ≤ck0(2k)! ,k=1, 2,. . ., for some constantc0>0. (f ) lim supk→∞21km12/(k2k)<∞.
ProofThe equivalence of (a), (b), (e) and (f ) was given in Theorem 1. To prove the remaining relations, denoteX∗ = |X| and write Yp = X∗p andνk,p=E Ypk
=μpk. Then note further thatEecX∗= E ec(Yp)1/p
constantc0 >0 iff lim supk→∞1kν 1/(pk)
k,p < ∞(by Lemma 2) iff (d) holds true. The proof is complete.
The above statements (e) and (f ) are special cases of (c) and (d) withp=2, respectively. Similarly, we give the following equivalence theorem without proof for Stieltjes case.
Equivalence Theorem B(Stieltjes case). Letp≥1 be a constant. Let the random vari-able 0 ≤ X ∼ F onR+with finitemk = μk = E[Xk] for all integersk ≥ 1. Then the following statements are equivalent:
(a)Xsatisfies Hardy’s condition, namely, the moment generating function of√Xexists. (b)μk≤ck0(2k)! ,k=1, 2,. . ., for some constantc0>0.
(c)μpk≤ck0(2pk+1),k=1, 2,. . ., for some constantc0>0. (d) lim supk→∞pk1μ1pk/(2pk)<∞.
(e) lim supk→∞1kμ1k/(2k)<∞.
3 Checkable criteria for moment indeterminacy
In this section we consider the checkable criteria for moment indeterminacy. Krein (1945) proved the following remarkable criterion in the Hamburger case.
Krein’s Theorem.LetX∼FonRhave a positive density functionfand finite moments of all positive orders. Assume further that the Lebesgue logarithmic integral
K[f]≡
∞
−∞
−logf(x)
1+x2 dx<∞. (1)
ThenFis M-indet onR.
We call the logarithmic integralK[f] in (1) the Krein integral for the densityf. Graffi and Grecchi (1978) as well as Slud (1993) proved independently the counterpart of Krein’s Theorem for the Stieltjes case by the method of symmetrization of a distribution onR+. To give a constructive and complete proof, we however need Lemma 3 below (see, e.g., Lin 1997, Theorem 3, and Rao et al. 2009, Remark 8).
Graffi, Grecchi and Slud’s Theorem.LetX∼FonR+have a positive density function f and finite moments of all positive orders. Assume further that the integral
K[f]=
∞
0
−logf(x2)
1+x2 dx<∞. (2)
ThenFis M-indet onR+and hence M-indet onR.
Lemma 3 Let Y have a symmetric distribution G with density g and finite moments of all positive orders. If the integral
K[g]=
∞
−∞
−logg(x)
1+x2 dx<∞,
then there exists a symmetric distribution G∗=G having the same moment sequence as G.
ProofBy the assumptions of the lemma, there exists a complex-valued functionφsuch that|φ| =g(in the sense of almost everywhere) and
∞
−∞φ(x)e
(see the proof of Theorem 1 in Lin 1997 for details, and Garnett 1981, p. 66, for the construction ofφ). The last equality implies that
∞
−∞x
kφ(x)eitxdx=0, t≥0,k=0, 1, 2,. . ..
In particular,
∞
−∞x
kφ(x)dx=0, k=0, 1, 2,. . ..
Letφ =φ1+iφ2, then bothφjare real and|φj| ≤g. We have ∞
−∞x
kφj(x)dx=0, j=1, 2, k=0, 1, 2,. . ..
We split the rest of the proof into three cases:
(i) φ1=0, φ2=0, (ii) φ1=0,φ2=0, and (iii) φ1=0, φ2=0.
(i) Ifφ1is odd, then for eacht>0, the functionφ∗(x):=φ1(x)sin(tx)is even and ∞
−∞x
kφ∗(x)dx=0, k=0, 1, 2,. . ..
Takeg∗=g+φ∗=g. Theng∗≥0 is even and has the same moment sequence asg. On the other hand, ifφ1is not odd, then let first(x)= 12[φ1(x)+φ1(−x)] which is even and satisfies
∞
−∞x
k(x)dx=0, k=0, 1, 2,. . ..
Next, takeg∗=g+=g, which has the same moment sequence asg. (ii) The proof of this case is similar to that of case (i).
(iii) If one φj is not odd, then it is done as in (i) (by taking(x) = 12[φj(x)+ φj(−x)] andg∗ = g+ ). Suppose now that both φj are odd, then, by the definition ofφ, we further have−∞∞ φ(x)eitxdx = 0∀t ∈ R. Lett > 0 be fixed and define the function ψ(x)=φ1(x)sin(tx)+φ2(x)cos(tx)(the imaginary part ofφ(x)eitx), then
∞
−∞x
kψ(x)dx=0, ∞ −∞x
kψ(−x)dx=0, k=0, 1, 2,. . ..
Takem(x)= 12[ψ(x)+ψ(−x)]=φ1(x)sin(tx)=0, which is even and satisfies ∞
−∞x
km(x)dx=0, k=0, 1, 2,. . ..
We haveg∗=g+m=g, which is nonnegative and has the same moment sequence asg. The proof is complete.
It should be noted that in the logarithmic integral (2), the argument of the density func-tionf isx2rather thanxas in (1). Recently, Pedersen (1998) improved Krein’s Theorem by the concept of positive lower uniform density sets and proved that it suffices to cal-culate the Krein integral over the two-sided tail of the density function (instead of the whole line).
Theorem 3(Pedersen 1998). Let X ∼ F on Rhave a density function f and finite moments of all positive orders. Assume further that the integral
K[f]=
|x|≥c
−logf(x)
1+x2 dx<∞for some c≥0. (3)
See also Hörfelt (2005) for Theorem 3 with a different proof (provided by H.L. Pedersen). Pedersen (1998) also showed by giving an example that Krein’s condition (1) is sufficient, but not necessary, for a distribution to be M-indet. This corrected the state-ment (2) in Leipnik (1981) about Krein’s condition. On the other hand, Pakes (2001) and Hörfelt (2005) pointed out the counterpart of Pedersen’s Theorem for the Stieltjes case. To prove this result, we need Lemma 4 below.
Theorem 4(Pakes 2001,Hörfelt 2005). Let X∼F onR+have a density function f and finite moments of all positive orders. Assume further that the integral
K[f]=
x≥c
−logf(x2)
1+x2 dx<∞for some c≥0. (4)
Then X is M-indet onR+and hence M-indet onR.
Lemma 4Let0 ≤ X ∼ F with density f and finite moments of all positive orders. Let Y ∼G with density g be the symmetrization of√X. If for some c≥0,
K[g]=
|x|≥c
−logg(x)
1+x2 dx<∞,
then X is M-indet onR+.
ProofUnder the condition on the logarithmic integral ofg, Pedersen (1998, Theorem 2.2) proved that the set of polynomials is not dense in L1(R,g(x)dx). This implies that the set of polynomials is not dense in L2(R,g(x)dx)either (see, e.g., Berg and Christensen 1981, or Goffman and Pedrick 2002, p. 162). Then proceeding along the same lines as in the proof of Corollary 1 in Slud (1993), we conclude that the set of polynomials is not dense in L2(R,f(x)dx). Therefore,Xis M-indet on R, which in turn implies thatXis M-indet onR+due to Chihara’s (1968) result in Fact A above. The proof is complete.
Conversely, once we prove Theorem 4, we can extend Lemma 4 as follows.
Lemma 4∗.IfX∼FonRsatisfies the conditions in Theorem 3, thenX2is M-indet.
ProofApply Theorem 4 above and Pakes et al.’s (2001) Theorem 3(i): IfX ∼ F onR satisfies condition (3), then the Krein integralK[f2] in (4) ofX2is finite, wheref2is the density ofX2.
For the M-det case, a trivial analogue of Lemma 4∗is the following.
Lemma 4∗∗. If X ∼ F on R satisfies Carleman’s condition (h7), then X2 satisfies Carleman’s condition (s6) and is M-det onR+.
For simplicity, all the conditions (1) through (4) are called Krein’s condition. For illustra-tion of how to use Krein’s and Hardy’s criteria, we now recover Berg’s (1988) results using these powerful criteria (see also Prohorov and Rozanov 1969, p. 167, Pakes and Khattree 1992, Lin and Huang 1997, and Stoyanov 2000).
Example 2Let X have a normal distribution andα >0.Then
(i) the odd power X2n+1is M-indet if n≥1,and (ii)|X|αis M-det iffα≤4. Without loss of generality, we assume that X has a density f(x)= √1
πexp
(I) Berg (1988) proved the moment indeterminacy of distributions by giving examples. For part (i), he calculated first the density of X2n+1:
fn(x)= 1
(2n+1)√π|x|
−2n/(2n+1)exp−|x|2/(2n+1), x∈R, and then constructed the density function
fr,n(x) = fn(x)
1+r cos
βn|x|2/(2n+1)
−γnsin
βn|x|2/(2n+1)
≡ fn(x){1+rpn(x)}, x∈R,
where|r| ≤ sin2(2n+π 1), βn = tan2n+π1 andγα = cot2(2n+π 1).It is seen that fr,n = fn if r = 0and n ≥ 1,but fr,n and fn have the same moment sequence because the product of the density fnand the function pn defined above has vanishing moments by a tedious calculation:
∞
−∞x
kfn(x)pn(x)dx=0,k=0, 1, 2,. . ..
This proves part (i). Alternatively, we note however that the Krein integral
K[fn]=
∞
−∞
−logfn(x)
1+x2 dx=C+ ∞
−∞
|x|2/(2n+1)
1+x2 dx<∞(if n≥1),
which implies by Krein’s Theorem that the odd power X2n+1is M-indet if n≥1. (II) For part (ii), the density of|X|αis
fα(x)= 2
α√πx1/α−1exp
−x2/α, x≥0. Ifα >4,Berg constructed again the density function
fr,α(x)=fα(x)
1+rcosβαx2/α−γαsinβαx2/α≡fα(x){1+rpα(x)}, x≥0, where|r| ≤sin(π/α), βα =tan(2π/α)andγα =cot(π/α).Then fr,α = fαif r =0and
α >4,but fr,αand fαhave the same moment sequence because
∞
0
xkfα(x)pα(x)dx=0, k=0, 1, 2,. . ..
Therefore,|X|αis M-indet ifα >4.Again, we see that the Krein integral (in Stieltjes case) K[fα]=
∞
0
−logfαx2
1+x2 dx=C+ ∞
0 x4/α
1+x2dx<∞(ifα >4).
So the required result follows immediately from Krein’s criterion (4). (III) For the rest of part (ii), Berg calculated the kth moment of|X|α:
mα,k =
∞
0
xkfα(x)dx= √1
π
αk+1 2
, k=0, 1, 2,. . ..
By Stirling’s formula, m1α/,kk ≈ ckα/2as k → ∞, and hence the Carleman quantity (in Stieltjes case) is equal to
C[fα]= ∞
k=1
mα−,1k/(2k)= ∞(ifα≤4).
There are some ramifications of the moment problem for normal random variables. For example, Slud (1993) investigated the moment problem for polynomial forms in normal random variables, while Hörfelt (2005) studied the moment problem for some Wiener functionals which extend Berg’s results in Example 2. Besides, Lin and Huang (1997) treated the double generalized Gamma (DGG) distribution as an extension of the nor-mal one and found the necessary and sufficient conditions for powers of DGG random variable to be M-det.
4 Stieltjes classes for M-indet distributions
Stieltjes (1894) observed that some positive measures, e.g.,μ(dx)=e−x1/4dxorxn−logxdx (n is an integer), are not unique by moments. This might be the starting point of T. J. Stieltjes to study the moment problem (see Kjeldsen 1993). It was C. C. Heyde who first presented this phenomenon in probability language and proved in 1963 that the lognormal distribution is M-indet by giving the example described next. Consider the standard lognormal density
f(x)= √1 2πx
−1exp
−1
2(logx) 2
, x>0,
with moment sequence{exp(k2/2)}∞k=1. Then, for eachε∈[−1, 1],
∞
0
f(x)[1+εsin(2πlogx)]xkdx=
∞
0
f(x)xkdx∀k=0, 1, 2,. . .
because the product of the density f(x) and the function sin(2πlogx) has vanishing moments:
∞
0
f(x)[sin(2πlogx)]xkdx= e k2/2 √
2π
∞
−∞e −x2/2
sin(2πx)dx=0∀k=0, 1, 2,. . ..
There are many other distributions having the same moment sequence as the above lognormal with mean √e, including (i) the ones with density f satisfying the func-tional equation: f(qx)=q−1/2xf(x), where q = 1/e ∈ (0, 1) (see, e.g., López-García 2011, Theorem 1), or, more generally, (ii) the distributions F satisfying F(x) = e−1/20exudF(u), x≥0 (Pakes 1996, Section 3). The latter showed that each suchF cor-responds to a finite measure in the interval(1/e, 1] and vice versa. Hence the cardinality of the set of all solutions to the functional equation isℵ2=2R. All these distributions are called the solutions to the lognormal moment problem (see also Chihara 1970, Leipnik 1982, Pakes 2007 and Christiansen 2003).
Recently, Stoyanov (2004) formulated Stieltjes classes for M-indet absolutely continuous distributions as follows. LetX ∼ Fhave an M-indet distribution onRwith densityf. A Stieltjes classSforFis defined by
S=S(f,p)= {fε:fε(x)=f(x)[1+εp(x)] ,x∈R, ε∈[−1, 1]},
wherepis a measurable function (called a perturbation function) such that|p(x)| ≤1 and
∞
−∞f(x)p(x)x
kdx=0,k=0, 1, 2,. . ..
1. IfX has a generalized Weibull densityf(x)= 241 exp(−x1/4), x>0,then
p(x)=sin(x1/4), x>0(Stieltjes 1894, Serfling 1980).
2. IfX has a density functionf(x)=cx−logx,x>0,wherec is a norming constant, then we choosep(x)=sin(2πlogx),x>0(Stieltjes 1894).
3. IfX has a gamma density with parameterα >0,thenXβis M-indet provided
β >max{2, 2α},and we can choose
p(x)=sin(απ/β)costanπ/β)x1/β−cot(απ/β)sin(tan(π/β)x1/β, x>0
(Targhetta 1990).
4. IfX has a density functionf(x)=cexp(−α|x|ρ),x∈R, whereα >0,ρ∈(0, 1)
andc is a norming constant, then we choosep(x)=cos(α|x|ρ), x∈R(Prohorov and Rozanov 1969, p. 167).
5. For the log-skew-normal distribution with parameterλ >0,we choose the perturbation function
p(x)= (x−1) (x)
sin[πlog(x−1)]
(λlogx) , ifx>1,
andp(x)=0,otherwise, whereis the density of standard lognormal LN(0,1) and
is the standard normal distribution (Lin and Stoyanov 2009).
Several systematic approaches for constructing Stieltjes classes are available. For exam-ple, for any M-indet distributionF on (0,∞) with density f bounded from below as f(x)≥Aexp(−αxβ),x>0, whereA>0, α >0 andβ ∈(0, 1/2)are constants, we find first a complex-valued functiongsatisfying
(i)gis analytic inC+\ {0}, whereC+= {z: Imz≥0}is the upper half-plane, and (ii)g(x)∈R,x>0, and|g(z)| ≤Aexp−α|z|β,z∈C+\ {0}.
Then choose the perturbation functionp(x)=[ Img(−x)]/f(x), x>0 (Ostrovska 2014). On the other hand, given any Stieltjes classS(f,p)defined above and a positive random variableVwith distributionHand finite moments of all positive orders, we can construct a new Stieltjes classS(f∗,p∗)by random scaling:Yε:=VXε,ε∈[−1, 1] , where the ran-dom variableXεhas densityfε,Vis independent ofXε,f∗=f0∗is the density ofY0=VX, and the perturbation functionp∗satisfiesf∗(x)p∗(x)=0∞v−1f(x/v)p(x/v)dH(v), x∈R (Pakes 2007, Section 5).
For more perturbation functions, see Stoyanov (2004), Stoyanov and Tolmatz (2004, 2005), Ostrovska and Stoyanov (2005), Gómez and López-García (2007), Penson et al. (2010), Wang (2012), Kleiber (2013, 2014) and Ostrovska (2016).
5 Converse criteria
In this section we present some converses to the previous M-(in)det criteria. Recall that for the Stieltjes case, if (s1) above holds true, i.e.,mk+1/mk = O((k+1)2)ask → ∞, thenXis M-det onR+. One might guess that if the moments{mk}∞k=1grow faster, then Xbecomes M-indet. This is true under one more condition defined below (see Lin 1997, Stirzaker 2015, p. 223, Kopanov and Stoyanov 2017, or Stoyanov and Kopanov 2017).
Lf(x):= − xf(x)
f(x) ∞ as x0<x→ ∞.
In the Hamburger case we require the densityf(x),x∈R, to be symmetric about zero.
Theorem 5Let X be a nonnegative random variable with distribution F and let its moments grow fast in the sense that mk+1/mk ≥c(k+1)2+ε for all large k, where c and εare positive constants. Assume further that X has a density function f which satisfies Condition L. Then X is M-indet.
Note that in the above theorem,Xis M-indet onR+iff it is M-indet onRbecauseX has a density. For the Hamburger case, we have the following.
Theorem 6Suppose the moments of X ∼ F on R grow fast in the sense that
m2(k+1)/m2k ≥c(k+1)2+εfor all large k, where c andεare positive constants. Assume fur-ther that X has a density function f which is symmetric about zero and satisfies Condition L. Then X satisfies Krein’s condition, and hence both X and X2are M-indet.
The crucial point in the proofs of Theorems 5 and 6 is to prove that the Krein integral K[f]< ∞by Condition L and the moment condition. The M-indet property ofX2in Theorem 6 is due to Lemma 4∗and Fact A above. Similarly, we have the following results for other criteria (s4) and (h5) (see Lin and Stoyanov 2015, 2016, and Stoyanov et al. 2014).
Theorem 7(Stieltjes case). Let X ∼F onR+and let its moments grow fast in the sense that mk≥c k(2+ε)k, k=1, 2,. . .,for some positive constants c andε. Assume further that X has a density function f which satisfies Condition L. Then X is M-indet.
Theorem 8(Hamburger case). Suppose the moments of X ∼ F grow fast in the sense that m2k ≥c(2k)(2+ε)k,k= 1, 2,. . .,for some positive constants c andε.Assume further that X has a density function f which is symmetric about zero and satisfies Condition L. Then X satisfies Krein’s condition, and hence both X and X2are M-indet.
Note that Condition L also applies to converse M-indet criteria. Actually, this is the original purpose of the condition, under whichK[f]= ∞impliesC[F]= ∞(Lin 1997). The M-det property ofX2in the next result is due to Lemma 4∗∗and Fact A above.
Theorem 9In Theorem3 (Hamburger case), if the Krein integral K[f]= ∞and if f satisfies Condition L, then X satisfies Carleman’s condition, and hence both X and X2 are M-det.
Theorem 10In Theorem4(Stieltjes case), if the Krein integral K[f]=∞and if f satisfies Condition L, then X satisfies Carleman’s condition and is M-det.
Remark 4In the above converse results, it is possible to replace Condition L by other slightly weaker conditions (mathematically) like those inPakes (2001)andGut (2002), but as mentioned before, we focus only on the checkable conditions in this survey. Interestingly, Condition L is closely related to a useful concept in reliability theory. More precisely, if a nonnegative random variable X with density F = f satisfies Condition L onR+with x0=0, then it has an increasing generalized failure rate (by Theorem 1 inLariviere 2006), namely, the product function xf(x)/F(x)(of x and the failure rate) increases in x.
In addition to the previous problems for normal distributions, we mention here some more variants for general cases, but we are not going to pursuit all the moment prob-lems. To solve these problems, we need to derive new auxiliary tools case by case (like Lemma 5 below). Lin and Stoyanov (2002) and Gut (2003) studied the moment prob-lem for random sums of independently identically distributed (i.i.d.) random variables. Stoyanov et al. (2014) and Lin and Stoyanov (2015) investigated the moment problem for products of i.i.d. random variables. In the next section we review the recent results about products of independent random variables withdifferentdistributions; for details, see Lin and Stoyanov (2016).
6 Moment problem for products of random variables
Products of random variables occur naturally in stochastic modelling of complex random phenomena in areas such as statistical physics, quantum theory, communication theory, reliability theory and financial modelling; especially in modern communications (see, e.g., Chen et al. 2012, Springer 1979, and Galambos and Simonelli 2004). We split the problem in question into three cases: (a) products of nonnegative random variables, (b) products of random variables taking values in R, and (c) the mixed case. Moreover, all random variables considered have finite moments of all positive orders.
6.1 Products of nonnegative random variables
The M-det result (Theorem 11 below) is an easy consequence of Theorem 2, while the hard part is the M-indet result (Theorem 12) whose proof needs a delicate analysis.
Theorem 11Letξ1,. . .,ξn be independent nonnegative random variables and let the moments mi,k=E ξik
, i=1,. . .,n,satisfy the conditions:
mi,k=O(kaik)as k→ ∞, for i=1,. . .,n,
where a1,. . .,an are positive constants. If the parameters ai are such thatni=1ai ≤ 2, then the product Zn=ni=1ξisatisfies Hardy’s condition and is M-det.
Theorem 12Consider n independent nonnegative random variables, ξi ∼ Fi, i = 1, 2,. . .,n, where n≥2.Suppose that each Fiis absolutely continuous and has a positive density fion(0,∞)and that the following conditions are satisfied:
(i) At least one of the densitiesf1(x),. . .,fn(x)is decreasing in[x0,∞),wherex0≥1is a constant.
(ii) For eachi=1, 2,. . .,n,there exists a constantAi>0such that the densityfiand the
tail functionFi(x)=1−Fi(x)=Pr(ξi>x)together satisfy the relation
and there exist constantsBi>0, αi>0,βi>0and realγisuch that
Fi(x)≥Bixγiexp
−αixβi for x≥x
0. (6)
If, in addition to the above,ni=11/βi>2,then the productZn=ni=1ξiis M-indet.
Let us explain the above conditions. In terms of reliability language, the failure rate in (5) and the survival function in (6) cannot approach zero too quickly. In other words, (5) and (6) control the tail (decreasing) behavior of the related distributions in some sense. There are three key steps in the proof of Theorem 12: (i) represent the density function of the productZnin multiple integral form, (ii) estimate the lower bound of the density function by truncating the two tails of this integral, and (iii) apply Krein’s criterion for the Stieltjes case. For estimation in the step (ii), we need the following auxiliary tool which can be proved using integration by parts.
Lemma 5Let F be a distribution onRsuch that (i) it has density f on the subset[a,ra] , where a>0and r>1,and (ii) for some constant A>0,f(x)/F(x)≥A/x on[a,ra] .Then
ra
a f(x)
x dx≥
1−1 r
A 1+A
F(a) a .
Example 3For illustration of how to use Theorems11and12, consider the generalized gamma distributions. We say thatξ ∼GG(α,β,γ )if its density is of the form
f(x)=c xγ−1exp(−αxβ), x≥0.
Hereα,β,γ > 0, f(0) = 0ifγ = 1,and c = βαγ /β/ (γ /β)is the norming constant. Then we have the following characterization result (see alsoPakes 2014for a much more general result with different proof ):
Suppose thatξ1,. . .,ξnare n independent random variables and letξi ∼GG(αi,βi,γi), i=1,. . .,n.Then the product Zn=ni=1ξiis M-det iff
n
i=11/βi≤2.
Example 4Consider the class of inverse Gaussian distributions. We say that ξ ∼ IG(μ,λ)if its density is of the form
f(x)=
λ
2πx3 1/2
exp
−λ(x−μ)2
2μ2x
, x>0,
whereμ,λ >0and f(0)=0.It can be shown that the product of two independent random variables is M-det if each one is exponential or inverse Gaussian, while the product of three such random variables is M-indet. For the powers of such random variables and others, see, e.g.,Lin and Huang (1997), Stoyanov (1999), Pakes et al. (2001), Stoyanov et al. (2014) andLin and Stoyanov (2015). Here are some recent results.
Letξ1 ∼ IG(μ1,λ1), ξ2 ∼ IG(μ2,λ2)andη ∼ Exp(1) = GG(1, 1, 1) be three inde-pendent random variables. Then both the productsξ1ηandξ1ξ2are M-det, whileξ1ξ2ηis M-indet.
6.2 Products of random variables taking values inR
Theorem 13Let ξ1,. . .,ξn be independent random variables and let the even order moments mi,2k =E ξi2k
, i=1,. . .,n,satisfy the conditions:
mi,2k =O((2k)2aik)as k→ ∞, for i=1,. . .,n,
where a1,. . .,an are positive constants. If the parameters ai are together such that
n
i=1ai≤1,then the product Zn=ni=1ξisatisfies Cramér’s condition and is M-det.
Theorem 14Consider n independent random variablesξi ∼ Fi,i = 1,. . .,n,where n≥ 2.Suppose each Fihas a positive density fionRand symmetric about zero. Assume further that
(i) at least one of the densitiesf1(x),. . .,fn(x)is decreasing in[x0,∞),wherex0≥1is a constant, and
(ii) for alli,fi/Fisatisfies the condition (5):fi(x)/Fi(x)≥Ai/x for x≥x0, andFisatisfies
the condition (6):Fi(x)≥Bixγiexp
−αixβi for x≥x 0.
If, in addition to the above,ni=11/βi >1,then the product Zn=ni=1ξisatisfies Krein’s condition, and hence both Znand Z2nare M-indet.
Example 5Applying Theorems13and14to the product of double generalized gamma random variablesξ ∼DGG(α,β,γ ),defined above, yields the following interesting result: Suppose that ξ1,. . .,ξn are n independent random variables, and let ξi ∼ DGG(αi,βi,γi), i=1, 2,. . .,n.Then the product Zn=ni=1ξiis M-det iff
n
i=11/βi≤1 iff Z2nis M-det.
6.3 The mixed case
Finally, we consider the products of both types of random variables, nonnegative and real ones taking values inR. Recall that this is the Hamburger case and the M-det cri-terion is similar to Theorem 13 and omitted. The next result about an M-indet cricri-terion extends slightly Theorem 5.1 of Lin and Stoyanov (2016). The proof is similar and is therefore omitted.
Theorem 15Consider n independent random variables divided into two groups. The first group,ξ1,. . .,ξn0,consists of nonnegative variables, while all the variables in the sec-ond group,ξn0+1,. . .,ξn,take values inR,where1 ≤ n0< n.Suppose that eachξi ∼ Fi has a density fiand that fi, i=1,. . .,n0,are positive on (0,∞),while fj, j=n0+1,. . .,n, are positive onRand symmetric about 0. Moreover, assume further that
(i) at least one of the densitiesfj(x),j=1, 2,. . .,n,is decreasing in[x0,∞),wherex0≥1 is a constant, and
(ii) for alli,fi/Fisatisfies the condition (5):fi(x)/Fi(x)≥Ai/x for x≥x0, andFisatisfies
the condition (6):Fi(x)≥Bixγiexp(−αixβi) for x≥x0.
If, in addition to the above,ni=11/βi >1,then the product Zn=ni=1ξisatisfies Krein’s condition, and hence both Znand Z2nare M-indet.
An application of the above theorem leads to the following interesting result:
Acknowledgements
The author would like to thank the Editor and two Referees for helpful comments and suggestions. Especially, one Referee pointed out the result in Lemma 4∗. The paper was presented at (1) the International Waseda Symposium, February 29 – March 3, 2016, held by Waseda University (Japan) and (2) the second International Conference on Statistical Distributions and Applications (ICOSDA), October 14–16, 2016, Niagara Falls, held by Central Michigan University (USA) and Brock University (Canada). The author thanks the organizers (1) Professor Masanobu Taniguchi and (2) Professors Felix Famoye, Carl Lee and Ejaz Ahmed for their kind invitations. The comments and suggestions of Professor Murad Taqqu and other audiences are also appreciated.
Competing interests
The author declares that there is no competing interest.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 2 March 2017 Accepted: 23 May 2017
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